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Theorem iooval2 13420

Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Assertion**
Ref	Expression
iooval2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem iooval2**
Step	Hyp	Ref	Expression
1		iooval 13411	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2		elioore 13417	. . . . . 6 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
3	2	ssriv 3987	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
4	1, 3	eqsstrrdi 4029	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ)
5		dfss2 3969	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ ↔ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
6	4, 5	sylib 218	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
7		inrab2 4317	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
8		ressxr 11305	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
9		sseqin2 4223	. . . . . 6 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
10	8, 9	mpbi 230	. . . . 5 ⊢ (ℝ^* ∩ ℝ) = ℝ
11	10	rabeqi 3450	. . . 4 ⊢ {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
12	7, 11	eqtri 2765	. . 3 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
13	6, 12	eqtr3di 2792	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
14	1, 13	eqtrd 2777	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 ℝ^*cxr 11294 < clt 11295 (,)cioo 13387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391

This theorem is referenced by: elioo2 13428 ioomax 13462 ioopos 13464 dfioo2 13490

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